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wrapped distribution : ウィキペディア英語版
wrapped distribution
In probability theory and directional statistics, a wrapped probability distribution is a continuous probability distribution that describes data points that lie on a unit ''n''-sphere. In one dimension, a wrapped distribution will consist of points on the unit circle.
Any probability density function (pdf) p(\phi) on the line can be "wrapped" around the circumference of a circle of unit radius. That is, the pdf of the wrapped variable
:\theta=\phi \mod 2\pi in some interval of length 2\pi
is
:
p_w(\theta)=\sum_^\infty .

which is a periodic sum of period 2\pi. The preferred interval is generally (-\pi<\theta\le\pi) for which \ln(e^)=\arg(e^)=\theta
==Theory==
In most situations, a process involving circular statistics produces angles (\phi) which lie in the interval from negative infinity to positive infinity, and are described by an "unwrapped" probability density function p(\phi). However, a measurement will yield a "measured" angle \theta which lies in some interval of length 2\pi (for example [0,2\pi)). In other words, a measurement cannot tell if the "true" angle \phi has been measured or whether a "wrapped" angle \phi+2\pi a has been measured where ''a'' is some unknown integer. That is:
:\theta=\phi+2\pi a.
If we wish to calculate the expected value of some function of the measured angle it will be:
:\langle f(\theta)\rangle=\int_^\infty p(\phi)f(\phi+2\pi a)d\phi.
We can express the integral as a sum of integrals over periods of 2\pi (e.g. 0 to 2\pi):
:\langle f(\theta)\rangle=\sum_^\infty \int_^ p(\phi)f(\phi+2\pi a)d\phi.
Changing the variable of integration to \theta'=\phi-2\pi k and exchanging the order of integration and summation, we have
:\langle f(\theta)\rangle= \int_0^ p_w(\theta')f(\theta'+2\pi a')d\theta'
where p_w(\theta') is the pdf of the "wrapped" distribution and ''a' '' is another unknown integer (a'=a+k). It can be seen that the unknown integer ''a' '' introduces an ambiguity into the expectation value of f(\theta). A particular instance of this problem is encountered when attempting to take the mean of a set of measured angles. If, instead of the measured angles, we introduce the parameter z=e^ it is seen that ''z'' has an unambiguous relationship to the "true" angle \phi since:
:z=e^=e^.
Calculating the expectation value of a function of ''z'' will yield unambiguous answers:
:\langle f(z)\rangle= \int_0^ p_w(\theta')f(e^)d\theta'
and it is for this reason that the ''z'' parameter is the preferred statistical variable to use in circular statistical analysis rather than the measured angles \theta. This suggests, and it is shown below, that the wrapped distribution function may itself be expressed as a function of ''z'' so that:
:\langle f(z)\rangle= \oint p_w(z)f(z)\,dz
where p_w(z) is defined such that p_w(\theta)\,d\theta=p_w(z)\,dz. This concept can be extended to the multivariate context by an extension of the simple sum to a number of F sums that cover all dimensions in the feature space:
:
p_w(\vec\theta)=\sum_^_F)}

where \mathbf_k=(0,\dots,0,1,0,\dots,0)^{\mathsf{T}} is the kth Euclidean basis vector.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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